Homework 1 – Probability Review 1. A class of 25 students is surveyed to find out how many own an MP3 player. Suggest a sample space to model all possible choices of the job scheduler. 2. A cell phone tower has a circular coverage area of radius 10km. We observe the source locations of calls received by the tower. (a) Suggest a sample space to model all possible source locations of calls that the tower can receive. (b) Using your sample space from part (a), what is the event that the source location of a call is between 2 and 5 km from the tower. 3. If P(C) and P(B ∩ C) are positive, derive the chain rule of conditional probability, P(A ∩ B│C) = (| ∩ )(|) Also show that P(A ∩ B ∩ C) = (| ∩ )(|)() 4. A certain jet aircraft’s autopilot has conditional probability 1/3 of failure given that it employs a faulty microprocessor chip. The autopilot has conditional probability 1/10 of failure given that it employs a nonfaulty chip. According to the chip manufacturer, the probability of a customer’s receiving a faculty chip is 1/4. Given that an autopilot failure has occurred, find the conditional probability that a faulty chip was used. Use the following notation: = { } = {ℎ }. 5. (a) If two sets A and B are disjoint, what equation must they satisfy? (b) If two events A and B are independent, what equation must they satisfy? (c) Suppose two events A and B are disjoint. Given conditions under which they are also independent. Give conditions under which they are not independent. 6. Each time you play the lottery, your probability of winning is . You play the lottery times, and plays are independent. How large should be to make the probability of winning at least once more that 1/2? 7. Given events A, B and C, show that P(A ∩ C│B) = (│)(|) Due date: September 16 Perfect score: 100 points (5 points per problem) if and only if P(A│B ∩ C) = (|) In this case, A and C are conditionally independent given B. 8. Let A and B be events for which P(A) , P(B) and P(A ∪ B) are known. Express the following in terms of these known probabilities: (a) P(A ∩ B). (b) P(A ∩ ). (c) P(B ∪ (A ∩ )). (d) P( ∩ ). 9. The infinite union bound. Show that for any infinite sequence of events , ( ⋃ ∞ = ) ≤ ∑ () ∞ = . 10. Let Ω : = {1,2,3,4,5}, A : = {1,2,3} and B : = {3,4,5}. Put P(A) : = 5 8 and P(B) : = 7 8 . (a) Find ℱ: = σ({A, B}), the smallest σ-field containing the sets A and B; (b) Compute P(F) for all F ∈ ℱ; (c) Trick question. What is P({1})? 11. Consider the sample space Ω := {−2, −1, 0, 1, 2, 3, 4}. For an event A ⊂ Ω, suppose that P(A)= |A|/|Ω|. For the random variable X (ω) := ω2 , find its probability mass function. 12. A class consists of 15 students. Each student has probability p = 0.1 of getting an “A” in the course. Find the probability that exactly one student receives an “A.” Assume the students’ grades are independent. 13. If X has mean m and variance 2, and if Y := cX , find the variance of Y . 14. A digital communication link has bit-error probability p. In a transmission of n bits, find the probability that k bits are received incorrectly, assuming bit errors occur independently. 15. Student heights range from 120 to 220 cm. To estimate the average height, determine how many students’ heights should be measured to make the sample mean within 0.25 cm of the true mean height with probability at least 0.9. Assume measurements are uncorrelated and have variance 2 = 1. What if you only want to be within 1 cm of the true mean height with probability at least 0.9? 16. Let X and Y be independent binomial (n, p) random variables. Find the conditional probability of X > k given that max (X ,Y ) > k if n = 100, p = 0.01, and k = 1. 17. Consider a probability space (S,F , P [.]). Suppose that D and E are events in F such that P [D] = 3/5 and P [E] = 4/5. From this information, is it possible to tell if D and E are mutually exclusive? Explain. 18. Let M be a positive integer and consider the sample space of outcomes S = F → R satisfies the three axioms of probability. Show that P : P (X = 1, Y = 2) = 1/6, P (X = 1, Y = 3) = 1/2, P (X = 2, Y = 2) = p, P (X = 2, Y = 3) = 1/6. (a) What is the value of p? Explain. (b) P (Y = 3) =? 20. 19. Suppose X and Y are random variables with joint probability mass function given by
